Dual Orlicz Mixed Quermassintegral

Abstract

We study the dual Orlicz mixed Quermassintegral. For arbitrary monotone continuous function \(\phi \), the dual Orlicz radial sum and dual Orlicz mixed Quermassintegral are introduced. Then the dual Orlicz–Minkowski inequality and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral are obtained. These inequalities are just the special cases of their \(L_p\) analogues (including cases \(-\infty<p<0\), \(p=0\), \(0<p<1\), \(p=1\), and \(1<p<+\infty \)). These inequalities for \(\phi =\log t\) are related to open problems including log-Minkowski problem and log-Brunn-Minkowski problem. Moreover, the equivalence of the dual Orlicz–Minkowski inequality for dual Orlicz mixed Quermassintegral and dual Orlicz–Brunn–Minkowski inequality for dual Orlicz mixed Quermassintegral is shown.